Determinant of the Schrödinger operator

For the example of the nonrelativistic Schrödinger operator, methods are formulated for calculating the determinant of an elliptic operator on the basis of scattering theory. It is shown that such a determinant is identical to the Jost determinant at zero energy. In the centrally symmetric case, it reduces to ordinary Jost functions and ultimately to the values of the zero-energy wave functions at the origin. The relationship between the determinant of the Schrödinger operator and the characteristics of the scattering resonances and the number of bound states in a field of opposite sign is noted. This makes it possible to find the first terms in the gradient expansion of the determinant as a functional of the potential. The problem of the correlation free energy of a classical plasma serves as a physical illustration.